Cosmology: Inflation, rate of expansion over the first few years

After inflation ended, there was a period of time when the universe was radiation-dominated. During this period the scale factor increased like $t^{1/2}$. As the expansion continued, the universe became matter-dominated and the expansion increased like $t^{2/3}$. If you plot either curve, you’ll see that they start out with infinite expansion rate and slow down. The slowing down is because gravity of both radiation and matter is attractive.

After billions of years of expansion, the density of radiation and matter got so low that the density of dark energy became important. This made the expansion start speeding up, because the gravity of dark energy is repulsive. This will eventually make the expansion go back to being exponential. But, unlike during inflation, the doubling time will be billions of years rather than a tiny fraction of a second.

The expansion of the universe is the just the motion of the matter in it. The universe continued to expand because of inertia. As in Newtonian physics, there's no force needed to make objects keep moving in general relativity.

Spacetime doesn't have a state of motion in general relativity. 3D space isn't particularly well defined to begin with, but to the extent that you can define it, it doesn't have a state of motion either. There's no way to define (in a generally covariant way) a notion of expanding space. The idea that the expansion of the universe is due to some special cosmology-only behavior of space is simply false. The next time someone tells you it's true, ask them to write down a tensor field that's nonzero when space is expanding and zero when it isn't, or quantitatively define in any other generally covariant way what they mean. They won't be able to do it.

The forces acting on the matter in the early universe were the inverse-square law of gravity and a repulsion due to the cosmological constant (which behaves like a force directly proportional to distance). The cosmological constant was too small to make a difference in the early (post-inflationary) universe. The inverse-square attraction slowed the rate of expansion, but not enough to stop it. The outcome of a small scale (and extremely homogeneous) explosion could be analyzed in the same way; spacetime would locally have a FLRW geometry, and you could put FLRW coordinates on it and define a "cosmological time" and "comoving distance" and such on that basis. There's no special magic in what happens at the cosmological scale; general relativity works the same at all scales.